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Fractal Music

Harlan Brothers with Benoit Mandelbrot at Yale[Click image to enlarge]

During an informal discussion in 2003 regarding the apparent lack of clarity associated with the subject, the late Benoit Mandelbrot suggested to Harlan Brothers that he undertake a mathematically rigorous treatment of fractal music. The first result was a lecture and lab that Harlan presented at the 2004 Fractal Geometry Summer Workshop at Yale. He has subsequently published several papers on the subject. "Structural Scaling in Bach’s Cello Suite No. 3," (Fractals Vol. 15, No. 1, 2007) reveals musical structure related to the Cantor set and helps to establish a mathematical foundation for the classification of fractal music. "Intervallic Scaling in the Bach Cello Suites," (Fractals Vol. 17, No. 4, 2009) describes a novel and robust approach to establishing the existence of power-laws in music.

His chapter entitled "The Fractal Nature of Music" will be appearing this spring in the World Scientific Publishing book "Benoit Mandelbrot: A Life in Many Dimensions." He also produced a compilation of fractal music in honor of Mandelbrot for Science Writers 2010, hosted at the Yale Peabody Museum. The program and dedication can be found here.

One of Harlan's early realizations was that musicians have been composing a form of fractal music for at least six centuries. Motivic scaling was familiar to many of the great Flemish composers such a Johannes Ockeghem and Josquin des Prez who developed the art of the mensuration or prolation canon. This type of canon is characterized by a melody or rhythmic motif that is repeated in different voices simultaneously at different tempos (here is an example). To be clear, not all mensuration canons are fractal; there are fundamental requirements that must be met in order to classify an object as such.

As with graphics, music can exhibit a wide variety of scaling behavior. In the course of exploring the role of power laws in music, Harlan has identified many types of scaling including, among others, self-similarity with respect to duration, pitch, interval, motif, and structure.
He has also written compositions to illustrate some of these scaling char-acteristics. The links below provide audio examples along with their descriptions (background information can be found here). While some of the compositions are too short to properly be considered fractal, they offer a hint of what one might listen for when searching for similar structure in the vast body of musical expression.

(Please note: Javascript must be enabled to listen to these compositions and read their descriptions.)